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The Tunneling Percolation Staircase Isaac Balberg Racah Institute of Physics, The Hebrew University, Jerusalem, Israel Stat. Mech. Day IV. 230611

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  • The Tunneling Percolation Staircase

    Isaac Balberg

    Racah Institute of Physics,

    The Hebrew University, Jerusalem, Israel

    Stat. Mech. Day IV. 230611

  • GRANULAR METAL COMPOSITES

    σ (x-xc)t

    t = 1.9

  • Thermal switches

    Airplane wheels

    Self regulated

    heaters

    Papyrus glyphs

    CB

    CB polymer composites and applications

  • Two fundamental questions

    1) What determines the conductivity threshold?

    2) What determines the shape of the σ(Φ) dependence?

    Phys. Rev. B, 30, 3933 (1984): 304 Q

  • Confirmation of universality

    0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.2410

    -4

    10-3

    10-2

    10-1

    100

    (

    cm

    )-1

    CB volume fraction

    t=1.95 , xc=0.10

    XC72 2011

    t = 1.7

    tun (D = 3) ≈ 2

    t

    cpp )(

  • Experimentally observed non universal exponents

    (p-pc)t

    t = 6.4

    vc = 39 vol.%

    p = v/vc

    R (≥ R0L1)

    RL= R(L/)/(L/)D-1

    R0

    (p-pc)-

    (p-pc)-(t-tun)

  • 5 10 15 20 25 30 35 400

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    r (arb.units)

    h(r

    ) or

    g(r

    )

    g(r) = g0exp[-2r/ξ]

    h(r) = (1/d)exp[-r/d]

    f(g) g(ξ/2d-1)

    The diverging normalized distribution

    in the tunneling percolation problem

    f(g) =h(r)(dr/dg)

    0gpc←p

    d

    ξ

    t = tun+(2d/ξ)-1Phys. Rev. Lett, 59, 1305 (1987): 215 Q

    σ (p-pc)t

  • Various carbon blacks

    t = 6.4 t = 4 t =1.8

    10 wt. %39 vol. %

  • Percolation and Tunneling?

    V V

    g(rij) = g0 exp[-2(rij-D)/ξ] What is the meaning of vc?

    Carbon, 40, 139 (2002): 100 Q.

  • The hopping model

    (the CPA network)

    rc

    Bc=(4π/3)[rc3- (2b)3]N

    The increase

    of N

    conserves

    topology but

    increase the

    value of g

    Local

    connectivity

    independent of N

    g(rc) = g0exp[-2(rc-2b)/ξ]

    Phys. Rev. B, 81, 155434 (2010)

  • The prediction of the percolation-threshold (critical-

    resistor, hopping like) model and its relation to the

    critical-percolation (phase-transition like) model

    ℓn(N)

    ℓn(σ)

    Δσc

    ℓn(σ) = -(aβ/ξ)Φ-β+ℓn(A2)

    Φ N1/3

  • The origin of the staircase model

    p

    Log(σ)

    pi,

    Zi

    g(rij) = g0exp[-2(rij)/ξ]

    J. Phys. D: 42, 064003 (2009)

    pc→ Bc/Z

  • The lattice and continuum

    composite models

    D

    ξ

  • The effect of tunneling in lattices and the continuum

    Phys. Rev. B, 82, 134201 (2010)

  • Staircase observed in N990/Polyethylene Composites

    0.45 0.50 0.55 0.60

    10-5

    10-4

    10-3

    10-2

    10-1

    (

    cm)-

    1

    CB volume fraction

    t=1.90 , xc=0.415

    t=1.70 , xc=0.465

    t=2.05 , xc=0.500

    t=1.95 , xc=0.570

  • Conclusions• The observed σ(Φ) behavior is a series of percolation

    transitions, the stairs.

    • In the lattice these have to do with the series of lattice neighbors. In the continuum these have to do with shells of neighbors that result from an RDF with peaks.

    • In the continuum the σ(Φ) behavior of each stair is that of a universal behavior while the envelope of the stairs yields a non universal percolation behavior which simply presents hopping.

    • The experimentally observed Φc is either of a “universal” stair or a result of the fact that the data do not go to Φ = 0. Percolation is well confirmed if both, a universal behavior and a non hopping behavior are exhibited by the data.

    • One can get a simple universal percolation behavior if all the resistors are nearly equal (non divergent distribution) and all belong to a random network (as for anisotropic particles).

  • The End

  • RF transmission (“AC Conductivity”) at 80 MHz

    0.45 0.50 0.55 0.6010

    -5

    10-4

    10-3

    10-2

    10-1

    100

    (

    cm

    )-1

    CB volume fraction

    t=0.4 , xc=0.415

    t=0.7 , xc=0.465

    t=1.65 , xc=0.500

    t=0.45 , xc=0.570

  • f(g

    )

    g

    Area = p = 2pcArea = pc

    What if we have a distribution of (or r0 = 1/g) values?

    RL (p-pc)-tun

    = gc∫1(1/g)f(g)dg.

    p[gc∫1f(g)dg] = pc

    f(g) = (1-)g- yields that:gc = [(p-pc)/p]

    1/(1-)

    (p-pc)-/(1-) (p-pc)

    -(t-tun)

  • Universal and Hopping Interpretations in

    XC72-Polyethylene Composite

    1.6 1.7 1.8 1.9 2.0 2.1 2.2

    10-4

    10-3

    10-2

    10-1

    100

    (

    cm

    )-1

    (CB volume fraction)-1/3

    , x-1/3

    (b)

    0.01 0.110

    -4

    10-3

    10-2

    10-1

    100

    (

    cm

    )-1

    CB , x-xc

    measured data

    t=1.95 , Xc=0.1

  • Non-Hopping but Percolation Behavior

    0.24 0.26 0.28 0.30 0.32 0.34

    10-10

    10-9

    10-8

    10-7

    10-6

    (Si content)-1 (vol. %)-1

    (Si content)-1/3 (vol. %)-1/3

    0.015 0.020 0.025 0.030 0.035 0.040 0.045

    Co

    nd

    uc

    tiv

    ity

    (

    cm

    )-1

    Phys. Rev. B, 83, 035318 (2011)

  • t

    cpp )(

    Percolation Transition

    t = tun = (D-2)+

    (1-p)

    tun (D = 2) ≈ 1.3

    tun (D = 3) ≈ 2

  • The LNB Model of the backbone

    R (≥ r0L1)RL= R (L/)/(L/)D-1

    RL (p-pc)-[ +(D-2))] (p-pc)

    -tun L1(p-pc)/pc =1 L1 (p-pc)-1

    (p-pc)-

    ζ ≥ 1

    r0L1 L

    S&A 1992

  • The electrical conductance in composites

    and the analysis of the conductance network

    One expects intuitively and we can show rigorously that

    V-Vc = (N-Nc)(vHC) (vol.%) in the continuum, is equivalent to ps-psc in lattices

  • TUNNELING (Wiesendanger, 1994)

    LL

    L ≤ 10 nm

    σ(r) = σ0exp[-2r/L]

    E

  • Resistors Equivalent Network

    (Ii) = (gij)(vi-vj)

    R = (v1-vN)/I

    I1 = -IN = I, Ii (i # 1,N) = 0

    Gii = Σgi,j, Gij = -gji

    (I) = G (v)

    (v) = G-1(I)

    Phys. Rev. B, 28, 3799 (1983): 162 Q

  • The critical behavior of the resistance of

    pencil generated line-segments system

    1/gl = A0l

    l

    J. Water Resources Research 29, 775 (1993) : 101 Q

  • Non Universal and Hopping Interpretations

    0.45 0.50 0.55 0.6010

    -6

    10-5

    10-4

    10-3

    10-2

    10-1

    100

    (

    cm)-

    1

    CB (volume fraction)

    t=10.95 , xc=0.305

    (a)

    1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34

    10-5

    10-4

    10-3

    10-2

    10-1

    (

    cm

    )-1

    (CB volume fraction)-1/3

    , x-1/3

    (b)

  • The model of Kogut and Straley

    f(g) = (1-)g-

    • =gc∫1 [f(g)/g]dg |(gc

    - -1)|, gc = [(p-pc)/p]1/(1-)

    • we have here the three types of = F(gc)behaviors; strongly-diverges for 0 < < 1, logarithmically diverges for = 0 and it will not diverge for < 0.

    • Correspondingly: (p-pc)-/(1-),

    • |log(p-pc)| and = [(1-)/(-)] = const.

  • The basics of non-universal behavior

    RL (p-pc)-tun

    = gc∫1(1/g)f(g)dg.

    p[gc∫1f(g)dg] = pc

    Example (K&S, 1979): f(g) = (1-)g- yields that:

    gc = [(p-pc)/p]1/(1-)

    (p-pc)-/(1-) (p-pc)

    -(t-tun)

    ( = [(1-)/(-)])

  • “RDF” of the polymer shells

  • “RDF” of the polymer shells

  • Evidence for stairs in MWCNT-polymer composites

  • Some idea about the materials

  • Evidence for stairs in MWCNT-polymer composites

  • Conductivity and viscosity

  • Conductivity and viscosity

  • RDF and Cumulative coordination number

  • RDF in the Continuum

  • Statistics of the t values in CNT-polymer

    composites

  • Two presentations of the same data

    Percolation interpretation “hopping” interpretation

  • Non Universal and Hopping Interpretations

    0.4 0.5 0.610

    -6

    10-5

    10-4

    10-3

    10-2

    10-1

    100

    Co

    nd

    uc

    tiv

    ity

    (c

    m)-

    1

    x (volume fraction)

    1.15 1.20 1.25 1.30 1.3510

    -6

    10-5

    10-4

    10-3

    10-2

    10-1

    100

    101

    Co

    nd

    uc

    tiv

    ity

    (c

    m)-

    1

    x-1/3

    (x volume fraction)

    -1.6 -1.4 -1.2 -1.0 -0.8 -0.610

    -6

    10-5

    10-4

    10-3

    10-2

    10-1

    100

    Co

    nd

    uc

    tiv

    ity

    (c

    m)-

    1

    log10

    (x - xC)

  • The physical and mathematical resemblance of the

    percolation and (CPA) hopping behaviors:

    In the classical hopping (CPA) model :log σ = -δc/ξ+log A1 = -(aβ/ξ)x

    -β +log A1 1/3 ≤ β ≤ 1

    In our percolation model: log σ = t log(x-xc) +log A2and we saw (from the Excluded volume argument) that

    t = tun+d/ξ -1 so that for t >> tun we have that t ≈ -(x-β)/ξ

    where 1/3 ≤ β ≤ 1 and log (x-xc) < 0.

    • The two models yield then quite a similar dependence on x. Experimentally; since usually x >> xc these two are usually non distinguishable. Can we still distinguish between them?

  • Presentation of the CPA results in the case of spheres

  • CB-polymer composites

    •Non universal value of t ≈ 6 for low structured CB composites

    (p-pc)t

    t = 6.4

    vc = 39 vol.%

    p = v/vc

    R (≥ r0L1)

    Phil. Mag. B, 56, 991 (1987): 142Q

    RL= R(L/)/(L/)D-1

  • Non Spherical Fillers

  • Graphene composites

  • effects of particle shape asymmetry: spheroids (ellipsoids of revolution)

    GTN for colloidal conductor-insulator composites

    prolate spheroid (a/b >1)

    oblate spheroid (a/b

  • σ(v) behaviors

    0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.910

    -7

    10-6

    10-5

    10-4

    10-3

    10-2

    v content

    2

    1

    Co

    nd

    uc

    tiv

    ity

    (

    cm

    )-1

    Ge film

    Ni-SiO2

  • Typical behavior of all composites

    Percolation Threshold

  • Non Universal behavior

  • Geometry of Local Resistors

    RV

    IRV-like

    (microemulsions)

    PT

    Local

    connectivity

    criterion?

    IRV

    (porous media,

    impurities in SC

    model)

  • A percolation threshold (critical resistor)

    system vs. a critical percolation (phase

    transition like) system

    ℓn(N)

    ℓn(σ)

    Δσc

    log σ = -(aβ/ξ)x-β+log A2

  • Critical Path AnalysisOne labels the conductances (the interparticle

    distances) in the system in descending order

    until one gets percolation i.e., one finds the

    thinnest soft shell δ/2 that still provides

    percolation for a given Φ.

    Actually, we implant hard core particles and

    then attach to each a shell of thickness δ/2.

    We check then how many particles, or how

    high a Φ is needed in order to get a

    percolation path.

    The agreement of the CPA with simulation of

    the resistors network suggests that one finds

    the largest σ(Φ) without having to calculate

    the entire resistor networks.

    This works because of

    the very wide resistor

    distribution values

  • The effect of tunneling in lattices and the continuum

  • The tunneling percolation staircase

    p

    Log(σ)

    pi,

    Zi

    tun= 1.9

    J. Mod. Phys. B, 18, 2091 (2004): 52 Q

  • The tunneling percolation staircase

    p

    Log(σ)

    pi,

    Zi

    tun= 1.9

    Int. J. Mod. Phys. B, 18, 2091 (2004): 52 Q

  • Two interpretations of the full set of data points

  • LATTICE PERCOLATION

    ps

    pb

    S

    pc

    r0

    J. Water Resources Research 29, 775 (1993) : 101 Q

  • A typical continuum system

    No p here

    Local

    connectivity

    by partial

    overlap

  • The End

  • The Staircase in the AC measurements

    0.45 0.50 0.55 0.60

    108

    109

    1010

    Y A

    xis

    Title

    X Axis Title

    C

    t=0.4

    t=0.7

    t=1.65

    t=0.45

  • Fluctuations in “experimental “ data

  • Two presentations of the same data

  • Two presentations of the same data

  • The volume and excluded volume

    of the capped cylinders

    v = (4/3)(W/2)3+(W/2)2L

    = (32/3)(W/2)3+8π(W/2)2L+4(W/2)L2

    = /4

    Nc = (Bc/v)v/ (Bc/v)(W/L)

    when L

  • Electro-Thermal Switching: the „global‟ effect

    • an increase of the macroscopic resistance above a threshold bias

    • a stationary “switched” state is reached within 30 sec

    -20 -10 0 10 20-0.3

    -0.2

    -0.1

    0.0

    0.1

    0.2

    0.3

    Cu

    rre

    nt

    (A)

    Voltage (V)

    t~1 sec

    t=30 and t=300 sec

    Phys. Rev. Lett. 90, 236601 (2003): 20 Q.

    V=3

    V

    V=6 V V=8 V

  • Making the Audio Disc

  • HISTORY

    • 1982 RCA Laboratories, Princeton, USA

    • Request: High conductivity, High elasticity.

    • Question: Carbon Black supplier?

    • A) Cabot Corporation, Boston USA

    • B) Akzo Chemicals, Amersfoort, The Netherlands

  • Low structure or high structure ?

  • Video Disc Radius (In)

    The “lost” signal (Percolation Threshold and Anisotropy)

    Phil Mag. B, 56, 991 (1987): 142 Q.

  • Overcoming the “Anisotropy”

    The Question is:

    How to express

    conceptually and

    mathematically the

    feeling that

    elongated, not

    parallel, particles will

    yield a higher

    conductivity (i.e.

    connectivity) for the

    same CB content?

  • Cylinder like objects

    Phys. Rev. B, 28, 3799 (1983): 156 Q

  • Behavior of three types of composites that are

    based on elongated carbon particles

  • Typical behavior of all composites

    Percolation Threshold

  • Typical behavior of all composites

    Percolation Threshold

  • the percolation picture implies the presence of a sharp cut-off, but tunneling does not

    percolation versus tunneling

    g(r12) = 0 for r12 > D+d

    percolation

    g(r12) ≠ 0 for r12 < D +d

    tunneling

    g(r12) = g0 exp[-2(r12-D)/]

    tunneling does not

    imply any sharp cut-

    off

    Carbon, 40, 139 (2002): 82 Q.

  • The concept of “excluded volume”

    R

    R

    2R

    Vex = v2D

    The Excluded

    Volume is the

    volume in which

    the center of the

    “other object”

    has to be in order

    for the two

    objects to, at least

    partially, overlap

    vNc = (v/Vex)(VexNc) = (v/Vex)Bc

    We generalized v2D to other objects calling it “Excluded Volume”

    Note: 2D = smallest Vex/v

    for permeable objectsPhys. Rev. B, 30, 3933 (1984): 290 Q

    NVex = B: the average

    number of overlapping

    objects, i.e. the average

    number of bonds per

    object. Bc is expected

    then to be a topological

    invariant.

  • History of percolation Thresholds in

    the Continuum (Sastry, PRE 2007)

    Table 2. Key percolation studies in three dimensional random fibrous materials

    Authors Year Reference Arrangement Objects Contribution Approach

    Balberg, Binenbaum and

    Wagner

    1984 27 isotropic capped cylinders

    First study of percolation of random objects in three

    dimensions

    Monte Carlo

    simulations

    Balberg, Anderson, Alexander et al.

    1984 26 isotropic capped cylinders

    Presented the conjecture that percolation threshold for a system of identical objects in three dimensions is inversely

    proportional to excluded volume of one object

    Monte Carlo simulations

    Bug, Safran and

    Webman 1985 28 isotropic

    capped

    cylinders

    Confirmed excluded volume

    rule and showed constant of proportionality equals one in slender rod limit

    cluster

    expansion

    Neda, Florian and Brechet

    1999 29 isotropic capped cylinders

    Performed Monte Carlo simulations of 3D stick systems using correct isotropic

    distribution and confirmed excluded volume theory

    Monte Carlo simulations

    Yi and Sastry 2004 30 isotropic ellipsoids

    Presented analytical

    approximation for the percolation threshold of two- and three- dimensional arrays of overlapping ellipsoids

    series

    expansion

    Yi, Wang, and Sastry 2004 31

    Uniform distribution in x-y plane. Range of

    distributions from parallel to random in z-direction

    ellipsoids

    Provided comparisons of cluster sizes, densities, and percolation points for two- and three-dimensional systems of

    overlapping ellipsoids to investigate the range of applicability of a 2D model for predicting percolation in thin

    three-dimensional systems

    Monte Carlo simulations

  • The generalization to the hard core caseIn considering

    tunneling, the

    tunneling range is

    R-r = δ

    The important

    observation is that :

    Фc = Ncv (Bc/ΔVex)v

    r2L/[L2(R-r)]

    [r/(R-r)](r/L)

    If δ = (R-r) we

    have that

    Фc (r/δ)(r/L)

    For L >> R

    Vex ≈ 4L2R

    ΔVex ≈ 4L2δ

  • For Oblate Particles

    Asymptotically for a >> b, d and isotropic distribution:

    ΔVex [(a+d)3 –a3] ≈ (a2d), v = πa2b, Фc b/d = (a/d)(b/a)

    From the simulations:

    Фc v/Vex ≈ (a/d)(b/a)1/2

    Phys. Rev. B, 81,155434 (2010).

    For discs of

    radius a and

    thickness b

    Vex a3

  • Resistors Equivalent Network

  • The tunneling percolation staircase

    p

    Log(σ)

    pi,

    Zi

    tun= 1.9

  • f(g

    )

    g

  • The RDF

  • The effect of a tunneling like distribution of conductances

    t = 6.4f(g) = (1-)g-

    (p-pc)-/(1-)

    f(g) g(ξ/d-1)

    t = tun+(d/ξ)-1

    t

    cpp )( V V

  • Resistors Equivalent Network

  • f(g

    )

    g

  • f(g

    )

    g

  • A percolation threshold (critical resistor)

    system vs. a critical percolation (phase

    transition like) system

    ℓn(N)

    ℓn(σ)

    Δσc

    log σ = -(aβ/ξ)x-β+log A2

  • Critical Path AnalysisOne labels the conductances in the system in

    descending order until one gets percolation

    i.e., one finds the largest soft shell δ that is

    needed to get percolation for a given Φ.

    Actually, one implants hard core particles and

    then attaches with a shell δ/2. One checks

    then how many particles, or how high a Φ is

    needed in order to get percolation.

    The agreement of the CPA with simulation of

    the resistors network suggests that one finds

    the largest σ(Φ) without having to calculate

    resistor networks.This works

    because of the

    very wide

    distribution of

    resistor values

  • Presentation of the CPA results in terms of percolation

    0.0 0.1 0.2 0.3 0.40

    10

    20

    30

    40

    50

    60

    Threshold Content C

    Th

    e e

    xp

    on

    en

    t

    t

  • application of CPA formulas to real conductor-polymer nanocomposites

    GTN for colloidal conductor-insulator composites

    from the knowledge of vs f and from the filler geometry [a/b and

    D=2max(a,b)] one can estimate the value of the tunneling factor

  • The physical and mathematical resemblance of the

    percolation and (CPA) hopping behaviors:

    In the classical hopping (CPA) model :log σ = -d/ξ = -(aβ/ξ)x

    -β +log A1 1/3 ≤ β ≤ 1

    In our percolation model: log σ = t log(x-xc) +log A2we saw (from the Excluded volume argument) that

    t = tun+d/ξ -1 so that for t >> tun we have that t ≈ -x-β/ξ

    where 1/3 ≤ β ≤ 1 and log (x-xc) < 0.

    • The two dependence yield then quite a similar dependence on x. Experimentally; since usually x >> xc these two are usually non distinguishable. Can we still distinguish between them?

  • application of CPA formulas to real conductor-polymer nanocomposites

    GTN for colloidal conductor-insulator composites

    even if a/b changes over 6 orders of magnitude, the tunneling factor is found to be within the

    expected range 0.1 nm < < 10 nm [G. Ambrosetti et al. Phys. Rev. B 81, 155434 (2010)]

  • The effect of tunneling in lattices and the continuum

  • Analysis of a resistors network

  • Critical Path AnalysisOne labels the conductances in the system in

    descending order until one gets percolation.

    One takes then the smallest conductance thus found

    and determines its δ(Φ) value. The agreement

    with simulation suggests that:

    One can simply arrange the δ(Φ) values and

    find directly the largest needed δ(Φ)

    without having to calculate resistor

    networks.

    This works because of the very wide

    distribution of resistor values

  • The excluded volume approach• The parameter that is known experimentally is the critical volume % of the “filler” Φc, so

    the prediction to be made is how does Φc depend on the microscopic parameters.

    • For spheres these parameters are D and so:

    • v= (4π/3)(D/2)3, Φ = Nv, Φc = Ncv,

    • Bc = (4π/3)(rc3-D3)N and δc = rc-D

    • Case 1: rc >> D → Bc = (4π/3)(δc3)N = (4π/3)(δc

    3)Φ/v

    • → δc (Φ/v)-1/3 →

    • Log ρ = α1 + β1Φ-1/3.

    • Case 2: rc ≈ D → Bc ≈ (4π/3)(3δcD2)N = (4π/3)(3δcD

    3)N/D = (3δc8)Φ/D

    • Φ/v → Bc = 24 δcΦ/D → δc = BcD/24Φ

    • δc DΦ-1

    • Log ρ = α1 + β1Φ-1.

  • The predicted CPA conductivity

    dependence on Φ

    • For capped cylinders:

    • v ≈ πR2L, Φ = Nv = πR2LN

    • ΔVex ≈ πL2δc, NΔVex = Bc

    • δc ≈ BcR[(R/L)]Φ-1 Φ-1

    • In general then, δc Φ-β where in 3D: 1/3 ≤ β ≤ 1

  • CPA for spheres

  • The RDF

  • The End

  • VISCOSITY vs. CONDUCTIVITY

  • The differences between the different

    types of elongated Carbon particles

    For high structure

    carbon black the

    aspect ratio is of the

    order of 10

  • The End

  • Recent results of the RF conductivity and

    permitivity of Carbon Black composites

  • -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.47.5

    8.0

    8.5

    9.0

    9.5

    10.0

    10.5

    11.0

    11.5

    measured 0.1GHz

    measured 1GHz

    measured 10GHz

    calculated 0.1GHz

    calculated 1GHz

    calculated 10GHz

    log

    10(

    (s

    ec

    -1))

    log10

    ((p-pc)/p

    c)

  • -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.40.8

    1.2

    1.6

    2.0

    2.4

    2.8

    3.2

    measured 0.1GHz

    measured 1GHz

    measured 10GHz

    calculated 0.1GHz

    calculated 1GHz

    calculated 10GHz

    log

    10(

    )

    log10

    ((p-pc)/p

    c)

  • Conductivity exponent

    0 2 4 6 8 10 12 14 16 18

    1.0

    1.5

    2.0

    2.5

    3.0

    3.5

    t

    f (GHz)

    N990

    Raven

    XC72

  • Permitivity exponent

    0 2 4 6 8 10 12 14 16 180.0

    0.4

    0.8

    1.2

    1.6

    2.0

    N990

    Raven

    XC72

    s

    f (GHz)

  • f(g

    )

    g

  • An illustration of the basic answer

    Occupied sites

    Empty sites

    If resistors R3 >> R2 >> R1 are attached, the resultant resistance of

    the system will be dominated by R2, i.e., for a given p, an R2 resistor

    will yield a threshold resistance of the system (R1→ 0 and R3 → ∞).

    Approach: Keeping p and going from Z1 to Z2 in the square lattice

    Bc = zpc =pzc

  • The staircase in a lattice

    p

    Log(σ)

    pi, Zi

  • The change of the average local

    resistance upon threshold approach

    (the bypassing network)

    • 1) Start from p > pc• 2) Consider the g‟s in descending order

    • 3) The largest pc of them yield percolation

    • 4) The smallest g in this subset is gc.

    • 5) If there is a distribution, gc decreases as p → pc.

    • 6) At p = pc the resistor of smallest g must be included

    • 7) If it is g → 0 it may result that → ∞

    UPON p→pc NOT ONLY THE NETWORK SHRINKS BUT THE AVERAGE RESISTANCE INCREASES AND MAY DIVERGE

  • A typical (pencil generated)

    continuum system

    No p

    no f.

    and N is not

    informative

    Phys. Rev. B, 28, 3799 (1983): 156 Q

  • The lattice composite model

  • Statistics and conductance of a composite system

  • RDF of spherical particles

  • Staircase structure in nano Graphite composites

  • Non Universal and Hopping Interpretations

    0.45 0.50 0.55 0.6010

    -6

    10-5

    10-4

    10-3

    10-2

    10-1

    100

    (

    cm)-

    1

    CB (volume fraction)

    t=10.95 , xc=0.305

    (a)

    1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34

    10-5

    10-4

    10-3

    10-2

    10-1

    (

    cm

    )-1

    (CB volume fraction)-1/3

    , x-1/3

    (b)

  • Ni-SiO2

    10-4

    10-3

    10-2

    10-1

    100

    0.20 0.30 0.40 0.50

    10-4

    10-3

    10-2

    10-1

    100

    Co

    nd

    uc

    tiv

    ity

    (c

    m)-

    1

    x (volume fraction)

    t = 3.3

  • Two fundamental questions

    1) What determines the

    percolation threshold?

    2) What determines the

    shape of the σ(x)

    dependence?

  • Another a priori unexpected behaviors

    0.4 0.5 0.6 0.7 0.8 0.910

    -1

    100

    101

    102

    vol. fraction (v)

    2

    1

    Co

    nd

    uc

    tiv

    ity

    (

    cm

    )-1

    fractional Ni content

  • Percolation and tunneling -percolation

  • Various carbon blacks

  • Two Conclusions The understanding of the conductivity dependence

    on the content of the carbon filler in terms of

    percolation theory is based on:

    σ (Φ-Φc)t

    • 1) The larger the aspect ratio and the isotropy

    the smaller the percolation threshold (the

    smaller the content of carbon particles needed

    for the onset of large conductivity), the

    “excluded volume” argument.

    • 2) The Behavior of the conductivity is

    determined by the distribution of the tunneling

    conductances (or interparticle distances) which

    is manifested by the deviation of the critical

    exponent t from its universal value.

  • The origin of the staircase model

    p

    Log(σ)

    pi,

    Zi

    g(rij) = g0exp[-2(rij)/ξ]

    J. Phys. D: 42, 064003 (2009)

    pc→ Bc/Z

  • The NLB Model

    A model for the links

    R (≥ r0L1)RL= R(L/)/(L/)D-1

    (p-pc)-[ +(D-2))] (p-pc)

    -tun(L1 (p-pc)

    -1)

  • An illustration of a Graphene composite

    v = πR2d

    Vex = π2R3

    v/Vex d/R

  • The Graphene-composite problem

    • For example, for high density spheres

    • δc = BcD/24,

    • for cylinders δc ≈ BcR[(R/L)]Φ-1 and

    • for discs Bc = π2(3b2δ+3bδ2+δ3)N so that

    • δ/t ≈ Bc/(3πФ), or ≈ Ф ≈ [t/5δ)], or . δ ≈ tBc/(3πФ),